How to extrude surface keeping face angles
As far as I know it can't be done with extrusion directly, but it can be approximated using the Move Along Normals operator.

Select the desired face and press G,G (G key twice) to initiate the Move Along Normals operator.

Move your face "backwards" towards the existing geometry so it slides along existing edges and Blender picks up the correct directions. Once along the desired edges, while moving, press C to unclamp movement.

Once unclamped you can now freely move "forward" towards opposite direction, while still keeping the alignment of the previous geometry.



The downsides of this method are that you won't be able to move in absolute distances in Blender units, only in percentage increments, and there won't be an edge left at the original position, if you did want to keep one there.